From Bradley Albrecht comes an ecologically minded enigma:
You are responsible for setting the ranger schedule at Riddler River National Park. Four rangers are assigned to two locations: the mountain lookout in the north and the lakeside campground in the south. Each assignment lasts one week (Monday through Friday), and every week two rangers should be in the north and two should be in the south.
Your task is to set an assignment schedule that lasts a certain number of weeks and then repeats indefinitely.
In the spirit of fairness, the rangers propose the following conditions for the schedule:
What is the shortest possible repeating schedule that meets the rangers’ conditions?
Since there are two locations, the first condition implies that the schedule is a multiple of 2 weeks.
Since there are three pairs of pairs (if Rangers A & B are paired, it is understood C & D are paired), the second condition implies that the schedule is a multiple of 3 weeks.
If there is a 6 week schedule,
$$\dfrac{6 \text{ weeks} \cdot 2 \text{ switches}}{4 \text{ rangers}} = 3 \text{ switches per ranger}$$This would contradict the second part of the third condition, as an even number of switches is necessary for each ranger to return to their starting assignment.
Note that one could also think of there being four positions instead of only two locations. For each location, exactly one of the rangers will 'Stay' the following week, and the other will 'Switch' to the other location.
Thus, 12 weeks does work.
The table below is one such schedule.
| Week | North | South | ||
|---|---|---|---|---|
| Stay | Switch | Stay | Switch | |
| 1 | A | B | C | D |
| 2 | A | D | B | C |
| 3 | C | A | D | B |
| 4 | B | C | D | A |
| 5 | B | A | C | D |
| 6 | D | B | A | C |
| 7 | C | D | A | B |
| 8 | C | B | D | A |
| 9 | A | C | B | D |
| 10 | D | A | B | C |
| 11 | D | C | A | B |
| 12 | B | D | C | A |
Note this schedule gives each ranger the same pattern of locations and moves, offset by 3 weeks. The pattern is: